Math 101 Intermediate Algebra
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Dividing and Simplifying Radicals
Chapter 8, Section 4


Remember that for the rest of this chapter, it will be assumed that all variables represent non-negative real numbers. So, for example

 =  a, a    0.


A Radical Expression Is Simplified When the Following Are All True

  1. No perfect powers are factors of the radicand.
  2. No radicand contains a fraction.
  3. No denominator contains a radical.

Statement 1 is accomplished by simplifying radicals as was done in section 3 of this chapter.

Statement 2 is accomplished by using the quotient rule and then rationalizing the denominator (see rationalizing the denominator and conjugates, below).

Statement 3 is accomplished by rationalizing the denominator (see rationalizing the denominator and conjugates, below).


Quotient Rule For Radicals
For non-negative real numbers a and b, b    0,


Rationalizing the Denominator

When the denominator of a fraction contains a radical, simplify the expression by

  1. Rationalizing the denominator (discussed next), or
  2. Using conjugates to rationalize the denominator (discussed below).

To rationalize a denominator, multiply both the numerator and denominator by a radical that will result in the radicand in the denominator becoming a perfect power.

Examples of rationalizing the denominator:


Using Conjugates to Rationalize the Denominator

When the denominator of an expression is a binomial with at least one of the terms a square root, then the denominator can be rationalized.

Do this by multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of a binomial is a binomial having the same two terms with the sign inbetween changed.

Binomial Expression Conjugate

Observe that when conjugates are multiplied together, the square root disappears

Examples of using conjugates to rationalize the denominator: